Implied Question: One cannot prove this principle by showing that it takes a twelve cubit string to go around a three by three cubit square, while it takes a nine cubit string to go around circle with a diameter of three cubits, as any circle that has a three Tefach circumference is one Tefach wide, as stated in our Gemara. (Why isn't this valid proof of the principle that a square is one quarter bigger than a circle with the same width?)

Answer: We cannot bring proof based on a string going around (the square) being one quarter bigger than (a circle with) that width. Do you think that a circle with a diameter of four square cubits contains the same area as a square that is three cubits square since the string surrounding it (i.e. perimeter and circumference) is the same length?

Answer (cont.): When you split a square that is three square cubits into nine squares of one cubit squared each, you will have nine square cubits. However, a circle with a diameter of four square cubits must have at least twelve such square cubits.

Proof #1: In a four square cubit square there is sixteen square cubits. This square is only quarter greater than a similar circle. This means that the circle (with a diameter of four that can fit into this square) has twelve square cubits. Rather, it must be we cannot prove this from the string that measures circumference and perimeter.

Proof #2: This is also apparent (that the rule cannot be proven based on perimeter) because a strip of five cubits by one cubit has a perimeter of twelve cubits. However, if you break it into square cubits, it is only five square cubits. This is because measuring squares (or rectangles) causes a narrowing of the thread at the corners, as opposed to a circle where the thread is extended.

Explanation: If we want to prove that a square is larger than a similar circle by one quarter, we can prove it as follows. If one were to take a dot and encircle it over and over again with many threads until it would be one square Tefach in a circle, and afterwards one would cut the string from this dot downwards (meaning the bottom half of the circle), and after he would do so he would spread out all of the strings (both from the top half and the bottom half, as per the picture in Tosfos), he would find that each string is slightly longer than the previous string (that was closer to the dot) on both sides.

Explanation (cont.): This would continue until one would get to the uppermost string (i.e. the longest string) which is three Tefachim long. This is the outer string which encircled a square Tefach, as any circle that is one Tefach wide has a circumference of three Tefachim.

Explanation (cont.): We therefore find that these strings are in an order as follows (referring to the picture of Tosfos where the two triangles form one big wide triangle). There is a wider strip of half a Tefach that goes until the starting point (this refers to the middle of the triangle, or the side of each triangle which is back to back with the other triangle), and the string gets thinner and thinner until it becomes a dot.

Explanation (cont.): If you were to split these triangles in half down this width of a half Tefach, you will find two triangles, each being one and a half Tefachim long (on the long side) and one half Tefach wide, and the other side ends in a point of nothing. Now, combine both of these triangles by placing the length of one opposite the shorter side of the other (rectangular picture in Tosfos). This will provide you with a rectangle that is one and a half Tefachim on one side and a half Tefach on the other.

Explanation (cont.): If you split this rectangle into three strips, you will find that there are three strips of a square half Tefach (resulting from this circle that has the diameter of a Tefach). However, if you were to divide a square Tefach into four even parts, you will find four square half Tefachs. This proves that a square is one quarter bigger than a circle of a similar area.

Observation: This is not an exact statement. The Gemara was not being didactic, as it (the diagonal) is actually slightly more (than two fifths greater than the sides of the square).

Proof: If you will make a square of ten by ten and you will divide it lengthwise and widthwise, you will find four squares of five cubits square each. If you then divide each of these squares diagonally, resulting in a diamond shape in the large square (as per the picture of Tosfos), you will find that this inner square (that is at an angle formed by the diagonals) is fifty square cubits. This is because it is half of the original square (of ten by ten square cubits), and each five by five square was split in half diagonally.

Proof (cont.): If each diagonal was based on the rule that it should be two fifths longer than the sides of the square, the inner square should be seven by seven cubits. This is not half of the outer square, as a square that is seven by seven cubits only has forty nine strips of one square cubit (i.e. an area of forty nine cubits). It would be fitting for it to be fifty cubits, as it is half of the ten by ten square which equals one hundred strips of a square cubit!

Observation: The Gemara is trying to answer Rebbi Yochanan's opinion in this fashion for no good reason, as he clearly does count the places where people sit.

Proof: This is because we know that Rebbi Yochanan openly says in Eruvin (76a) that a round window needs to have a circumference of twenty four cubits. The only way this can be understood is according to the judges of Kisri etc.

Explanation #1: Rashi explains that when you make a square inside a circle, you take away half of what is left in the square, which means one third of the entire circle. Therefore, for a square of sixteen one requires a circle that surrounds it (i.e. circumference) to be twenty four.

Question: This is bewildering! What do these two things have to do with each other? (Even if the area of the circle would be one third more than that of the square, what does this have to do with the circumference of the circle, as Rashi focuses on the length surrounding the circle, and the perimeter of the square? The circumference of the circle, whose diameter is the diagonal of the square which is not more than six (based on the one and two fifths rule, see end of our Tosfos), is certainly not that much greater than the perimeter of the square! - based on Tosfos ha'Rosh)

Explanation #2: Rather, the explanation of the statement, "a square that is made inside a circle is half" refers to half the width of the circle being the same amount as the side of the square. This means that the corners of the square in the circle are twice as big as the width of the square, since he holds that any square cubit has a diagonal of two cubits.

Question: This is difficult. How did Rebbi Yochanan and the judges of Kisri make such a mistake in measurement? As they obviously did not measure (as it is clearly incorrect), how did they make a rule which is incorrect?

Explanation #3: It is possible to say that they had a tradition with this wording, "A square inside a circle is half." It is true regarding the area, just not regarding the circumference or perimeter.

Proof: This is clearly true, as if you make a square of ten by ten and divide it vertically and horizontally (four square of five by five), and you will then make a diamond within the large square as I stated earlier (Tosfos 8a DH "Kol," as per the picture in Tosfos both here and there), the inner diamond square is exactly half of the size of the large square.

Proof (cont.): If you will make a circle of ten by ten (i.e. with a diameter of ten) in the outer square, this circle will be between the inner (i.e. diamond) and outer square. We have proven (8a, DH "Kamah") that a square is one quarter more than a circle (if both have the same diameter or width straight across). If so, the circle is larger than the inner square by one half of the area of the inner square, which is one quarter of the outer square.

Question: It is somewhat difficult that the Gemara does not say that a square that is inside a circle is one third less than the circle, as it does say that a circle inside a square is one quarter less than the square. (It would seem that our statement in the Gemara should match the previous statement.)

Answer: The explanation therefore is that these two statements are indeed consistent. The case is as I have stated, that it is referring to a (smaller diamond) square within a circle that is within a (large) square. The Gemara is saying that the inner circle is one quarter less than the outer square, and that the inner square is one half of the outer square. It is therefore consistent, as they are both referring to their size relative to the outer square.

Question: If one were to make a circle inside a square that is seven by seven with a diameter of seven, and then they would draw a square inside the circle that diagonally would measure seven (i.e. a diamond) like the diameter of the circle, the inner square would be five by five based on the formula that a diagonal is one and two fifths of the width of a square. Accordingly, the inner square is more than half of the outer square, as the inner square is twenty five square cubits whereas the outer square is forty nine!

Answer: This depends on what we have proven previously (8a, DH "Kol") that the formula of one and two fifths is not exact, as it is actually slightly more. If so, the inner square is not actually five by five (rather it must be less), as if it would be its diagonal would be more than seven.

Explanation: A guard hut is obligated in Mezuzah, as the Beraisa states in Menachos (33b) that a guard hut and two different types of porches are obligated in Mezuzah.

Question: This is difficult, as in Yoma (11b) we say that we exclude such a hut from Mezuzah, as the Pasuk says, "house." We derive that a house indicates that to be obligated it must be specifically for living, as opposed to places that are not exclusively for living.

Answer: Some differentiate between whether these places are open to the house or to the garden.

Question: This is difficult. If this is the case, the Gemara should have answered in Menachos (ibid.) when it asked a question on Rav Chisda who says that an Achsadrah is exempt from Mezuzah.

Answer: One must say that one is according to Torah law and one is according to Rabbinic law. The Gemara in Menachos does not answer this, as it understands that Rav Chisda implies it is completely exempt, even according to Rabbinic law.

Explanation: All of these (people involved in the cases) stated in our Gemara are regarding people that do not have an obligation of Sukah. The Kusim do not have an obligation because this opinion understands that they converted out of fear (and therefore are not valid converts).

Implied Question: The Gemara in Yevamos (24b) states that both converts who converted out of fear and converts who convert because a person who interprets dream told them to do so are considered converts. (How can we say they are not valid converts?)

Answer: The Gemara in Yevamos (ibid.) refers to people who converted fully. However, the people who converted out of fear did not convert fully, as stated in Melachim (2, ch. 17) that they would fear Hash-m, but they would worship their idols. The opinion that holds they were true converts understands that they ended up only serving Hash-m.